Question

The heights of women aged 30 to 39 are approximately normal with mean 63 inches and standard deviation 2.5 inches. Men the same age have mean height 69.7 inches with standard deviation 2.7 inches. What are the z-scores (the standardized values, relative to the mean and standard deviation for the distribution of the correct sex) for a woman 6 feet tall and a man 6 feet tall? What information do the z-scores give that the actual heights do

Answer 1

The z-score for a woman who is 6 feet tall in the context given is 3.6, and for a man of the same height, it is approximately 0.852. These scores indicate how many standard deviations an individual's height is from the mean height of their gender group.

Step-by-step explanation:

To calculate the z-score for a woman who is 6 feet tall (72 inches), we'll use the formula:

z = (x - μ) / σ

For women: μ = 63 inches, σ = 2.5 inches.
So, z = (72 - 63) / 2.5 = 3.6.

For men: μ = 69.7 inches, σ = 2.7 inches.
So, z = (72 - 69.7) / 2.7 ≈ 0.852.

A z-score of 3.6 for the woman means her height is 3.6 standard deviations above the mean height for women. A z-score of 0.852 for the man indicates his height is 0.852 standard deviations above the mean height for men. Z-scores help compare individuals' heights to the average height of their respective groups, measured in units of standard deviation.